So I want to clarify/verify a specific solution. This is my solution to a problem, so it is not like I want to verify a specific proof/theorem or clarify why and/or how it works. So, in my tags, I write (solution-verification) but for some reason the tag shows as (proof-verification). Why is this? It only does this on the Math Stack Exchange as far as I know. I wanted to use the tag on the Math Overflow but I need at least $300$ reputation to use that tag, and I don't have that. Any thoughts?
This is is actually not a bug. When you look more closely at the tag (proof-verification) you can see that there is actually a tag synonym solution-verification $\to$ proof-verification with (proof-verification) as the master tag: https://math.stackexchange.com/tags/proof-verification/synonyms. It is a basic feature of tag synonyms that when you enter some of the synonyms, it is always changed to the master tag.
Whether or not they should be synonyms is separate question. (However, reverting synonyms is usually not simple. If the synonym is cancelled, still many of the question would have to be retagged manually; in the case of a tag containing 14k questions this would be a rather ambitious undertaking. For more details see: What happens with the tagged questions when a tag-synonym is cancelled?)
Whether or not these two tags should be synonyms was discussed here: What is the usefulness of having “proof-verification” and “solution-verification” as different tags? My impression is that no clear consensus was reached in that discussion. But a synonym was created despite the fact that the previous discussion on meta was -at least in my opinion inconclusive.